The conditions for TBM mixing [1]}, [2]}, [3]}, and its deviation to \(\rm {TM}_2\)  require the imposition of the following assumptionsThe validity of these assumptions is discussed in appendix D of Ref. [4]}. on \(m_{D}\)  and \(m_{M}\) : \(y_{1}=y_{2}\) , \(y_{3}\upsilon _{\rho _{1}}+y_{5}\upsilon _{\digamma }=2y_{4}\upsilon _{\rho _{1}}\) , and \(y_{8}\upsilon _{\rho _{3}}=-y_{7}\upsilon _{\rho _{2}}=y_{6}\upsilon _{\Gamma }/2 \) .
Now, we can apply the usual seesaw formula \(m_{\nu }=m_{D}m_{M}^{-1}m_{D}^{T}\)  which gives rise to the total neutrino mass matrix
\(m_{\nu }=\frac{m_{0}}{P}\left(\begin{array}{ccc}-\left( a+b\right) ^{2} & \left( a+b\right) \left( b+\mathrm {k}\right) &b^{2}-\mathrm {k}^{2}-b\left( \mathrm {k}-a\right) \\\left( a+b\right) \left( b+\mathrm {k}\right) & -\left( b+\mathrm {k}\right)^{2} & -a^{2}-ab+b^{2}+\mathrm {k}b \\b^{2}-\mathrm {k}^{2}-b\left( \mathrm {k}-a\right) & -a^{2}-ab+b^{2}+\mathrm {k}b & a\mathrm {k}-b^{2}\end{array}\right) \) 
